# Incremental Eigenpair Computation for Graph Laplacian Matrices: Theory and Applications

arXiv: https://arxiv.org/abs/1801.08196 · HTML: https://mlsanity.com/paper/1801.08196/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Dec 2017.

## Built on by (most cited first, 1 of 1)

- [Can Aggregate Invariants Accelerate Continuous Subgraph Matching? Limits, Laws, and a Dynamic Spectral Index](https://mlsanity.com/paper/2606.24421/) (2026)

## Abstract (arXiv)

The smallest eigenvalues and the associated eigenvectors (i.e., eigenpairs) of a graph Laplacian matrix have been widely used in spectral clustering and community detection. However, in real-life applications the number of clusters or communities (say, K) is generally unknown a-priori. Consequently, the majority of the existing methods either choose K heuristically or they repeat the clustering method with different choices of K and accept the best clustering result. The first option, more often, yields suboptimal result, while the second option is computationally expensive. In this work, we propose an incremental method for constructing the eigenspectrum of the graph Laplacian matrix. This method leverages the eigenstructure of graph Laplacian matrix to obtain the K-th smallest eigenpair of the Laplacian matrix given a collection of all previously computed K-1 smallest eigenpairs. Our proposed method adapts the Laplacian matrix such that the batch eigenvalue decomposition problem transforms into an efficient sequential leading eigenpair computation problem. As a practical application, we consider user-guided spectral clustering. Specifically, we demonstrate that users can utilize the proposed incremental method for effective eigenpair computation and for determining the desired number of clusters based on multiple clustering metrics.